Here we´ll elaborate how this problem can be solved by using 4-invariants for the 4-velocities of the spaceship. To begin with, let us consider the following 4-velocities:
(A) u – the 4-velocity of the laboratory frame;
(B) v – the 4-velocity of the spaceship after the first acceleration stage (i.e. at the moment \tau as measured in the spaceship’s proper time from the moment of launch);
(C) w – the 4-velocity of the spaceship after the second acceleration stage.
Based on these three 4-velocities, one can construct six independent 4-invariants, e.g. u_tv_t-u_xv_x-u_yv_y-u_zv_z. The fact that these expressions are invariant, i.e. yield the same result regardless of the reference frame used, makes it possible to obtain the components of u in the frame (C), as well as the components of w in the frame (A) by referring to the components u in the frames (A) and (B), v in the frames (A), (B) and (C), and w in the frames (B) and (C) (all of which we already know).
Please submit the solution to this problem via e-mail to physcs.cup@gmail.com. The next hint for the Problem 4 will be published at 13:00 GMT, 26th March 2023, together with the next updates of the intermediate results.